Travelling Waves of a Delayed SIR Epidemic Model with Nonlinear Incidence Rate and Spatial Diffusion

Abstract

This paper is concerned with the existence of travlelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.

Introduction

Let represent the number of individuals who are susceptible to the disease, that is, who are not yet infected at time ; represent the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals, and represent the number of individuals who have been infected and then removed from the possibility of being infected again or of spreading at time . In [1], Cooke formulated a SIR model with time delay effect by assuming that the force of infection at time t is given by , where is the average number of contacts per infective per day and is a fixed time during which the infectious agents develop in the vector, and it is only after that time that the infected vector can infect a susceptible human. Cooke considered the following model

(0.1)

where parameters , , are positive constants representing the death rates of susceptibles, infectives, and recovered, respectively. It is natural biologically to assume that . The parameters and are positive constants representing the birth rate of the population and the recovery rate of infectives, respectively. Much attention has been paid to the analysis of the stability of the disease free equilibrium and the endemic equilibrium of system (1.1) (see, for example, [2], [3], [4], [5]).

Incidence rate plays an important role in the modelling of epidemic dynamics. It has been suggested by several authors that the disease transmission process may have a nonlinear incidence rate. In [6], Xu and Ma considered the following SIR epidemic model with time delay and nonlinear incidence rate

(0.2)

By analyzing the corresponding characteristic equations, they studied the local stability of an endemic equilibrium and a disease free equilibrium. It was proved that if the basic reproductive number , the system was permanent. By comparison arguments, it was shown that if , the disease free equilibrium was globally asymptotically stable. If , by means of an iteration technique and Lyapunov functional technique, respectively, sufficient conditions were obtained for the global asymptotic stability of the endemic equilibrium.

We note that the spatial content of the environment has been ignored in the models aforementioned. The models have been traditionally formulated in relation to the time evolution of uniform population distributions in the habitat and are as such governed by ordinary differential equations. However, due to the large mobility of people within a country or even worldwide, spatially uniform models are not sufficient to give a realistic picture of disease diffusion. For this reason, the spatial effects can not be neglected in studying the spread of epidemics. Noble [7] applied reaction-diffusion theory to describe the spread of plague through Europe in the mid-14th century. By using the linear theory of semigroups, Saccomandi [8] investigated the existence and uniqueness of the solution for an SIR model with spatial inhomogeneity, nonlocal interactions, and an open population. In recent times, many investigators have introduced population movements into related equations for epidemiological modeling and simulations in efforts to understand the most basic features of spatially distributed interactions (see, for example, [9], [10], [11], [12], [13]).

Motivated by the work of Xu and Ma [6] and Noble [7], in the present paper, we are concerned with the following delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

(0.3)

with initial conditions

(0.4)

In problem (3)-(4), the positive constants , and denote the corresponding diffusion rates for the susceptible, infected and removed populations, respectively; is a bounded domain in with smooth boundary . The functions are nonnegative and Hölder continuous and satisfy in . In this paper, we assume that .

In the biological context, it is important to analyze the epidemic waves which is described by traveling wave solutions propagating with a certain speed. In this paper, we are interested in the existence of travelling wave solutions to SIRS epidemic model (0.3). The main tool to study the existence of travelling wave solutions for the reaction-diffusion equations with delays is the sub- and supersolution technique due to Atkinson and Reuter [14]. Wu and Zou [15], [16] studied the existence of travelling wavefronts for delayed reaction-diffusion systems with reaction terms satisfying the so-called quasi-monotonicity or exponential quasi-monotonicity conditions, where the well-known monotone iteration techniques for elliptic systems with advanced arguments [17], [18] were used. Ge and He [19] and Ge et al. [20] used the iteration technique developed by Wu and Zou [15] to investigate the existence of travelling wave solutions for two-species predator-prey system with diffusion terms and stage structure, respectively. However, we note that the nonlinear reaction terms of system (0.3) do not satisfy either the quasi-monotonicity or the exponential quasi-monotonicity conditions. Therefore, the method of upper-lower solutions and its associated monotone iteration scheme developed by Wu and Zou [15], [16] can not be used to study the existence of travelling wave solutions to system (0.3). Recently, by constructing a pair of desirable upper-lower solutions, Huang and Zou [21] got a subset, and employed the Schauder's fixed point theorem in this subset to investigate the existence of travelling wave solutions of a class of delayed reaction diffusion systems with two equations in which the nonlinear reaction terms satisfy the partial quasi-monotonicity and partial exponential quasi-monotonicity, respectively. Li et al. [22] investigated the existence of travelling wave solutions of a class of delayed reaction diffusion systems with two equations in which the reaction terms satisfy weak quasi-monotonicity and weak exponential quasi-monotonicity conditions, respectively. Sazonov et al. [23], [24] studied the problems of travelling waves in the SIR model. Clearly, all the results above can not directly be applied to a system with more than three equations. Therefore, it remains important and challenging to study the existence of travelling wave solutions for delayed reaction diffusion systems with more than three equations in which the nonlinear reaction terms do not satisfy either the quasi-monotonicity or the exponential quasi-monotonicity conditions.

Methods

Preliminaries

Throughout this paper, we adopt the usual notations for the standard ordering in . Thus, for and , we denote if , ; if but ; and if but , . If , we also denote , , and . We use to denote the Euclidean norm in and to denote the supremum norm in .

Before proving the existence of travelling wave solutions to system (0.3), we first investigate the following general delayed reaction-diffusion system:

(0.5)

On substituting , , and denote the travelling wave coordinate still by , we derive from (0.5) that

(0.6)

satisfying the following partial quasi-monotonicity conditions :

There exist three positive constants such that

(0.7)

where , with , , , , are positive constants, , , , and the functions are defined by

If, for some , system (5) has a solution defined on satisfying

(0.8)

where and are steady states of (0.5). Then , , is called a travelling wave solution of system (0.5) with speed . Without loss of generality, we can assume  = (0,0,0) and , and seek for travelling wave solution connecting these two steady states.

Corresponding to (0.5), we make the following hypotheses:

(A1) , .

(A2) There exist three positive constants such that

for , with , , , .

In the next section, we will apply the Schauder's fixed point theorem, which requires the continuity of the operator under consideration. For this purpose, we need to introduce a topology in . Let and equipped with the exponential decay norm defined by

Define

Then it is easy to check that is a Banach space.

We look for travelling wave solutions to system (0.5) in the following profile set:

Obviously, is non-empty, convex, closed and bounded.

We also need the following definition of upper and lower solutions to system (0.5).

Definition 0.1

A pair of continuous functions and are called a pair of upper-lower solutions of system (0.5) if and are twice differentiable almost everywhere in and they are essentially bounded on , and there hold

(0.9)

(0.10)

(0.11)

and

(0.12)

(0.13)

(0.14)

Unlike the standard upper and lower solutions defined in2₃ [15], is evaluated in a cross iteration scheme given in (0.10) and (0.13).

Local stability

In this section, by analyzing the corresponding characteristic equations, we discuss the local stability of an endemic equilibrium and a disease-free equilibrium of system (0.3) with the initial conditions (0.4) and the homogeneous Neumann boundary conditions

(0.15)

respectively, where denotes the outward normal derivative on , the homogeneous Neumann boundary conditions imply that the populations do not move across the boundary .

System (0.3) always has a disease-free steady state . Further, if , then system (0.3) has a unique endemic steady state , where

Let

is called the basic reproductive number (sometimes called basic reproductive rate or basic reproductive ratio) of system (0.3), which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process. It is easy to show that if , the endemic steady state exists; if , is not feasible.

Let be the eigenvalues of the operator on with the homogeneous Neumann boundary conditions, and be the eigenspace corresponding to in . Let , be an orthonormal basis of , and . Then

Let , , , where

and represents any feasible uniform steady state of system (0.3). The linearization of system (0.3) at is of the form . For each , is invariant under the operator , and is an eigenvalue of if and only if it is an eigenvalue of the matrix for some , in which case, there is an eigenvector in .

The characteristic equation of takes the form

(0.16)

where

Clearly, for any , Eq. (0.16) always has one negative real root . Its other roots are determined by the following equation

(0.17)

When , Eq. (0.17) becomes

(0.18)

It is readily seen that if , then

Hence, if , the endemic steady state of system (0.3) is locally stable when .

If is a solution of (0.16), separating real and imaginary parts, we derive that

(0.19)

Squaring and adding the two equations of (0.19), it follows that

(0.20)

Let , then Eq. (0.20) becomes

(0.21)

By calculation it follows that for all

Hence, we can know that Eq. (0.21) has no positive roots. Therefore, if , is locally asymptotically stable for all .

The characteristic equation of is of the form

(0.22)

Clearly, for any , Eq. (0.22) always has a negative real root . All other roots are given by the roots of equation

(0.23)

Let

If , it is readily seen that for real,

Hence, when , (23) has a positive real root. Therefore, there is a characteristic root with positive real part in the spectrum of . Accordingly, if , is unstable.

We therefore obtain the following results.

Theorem 0.1

For system (0.3) with initial conditions (0.4) and homogeneous Neumann boundary conditions (0.15), we have

(i) If , the disease-free steady state is unstable; if , is locally asymptotically stable.

(ii) Let , the endemic steady state is asymptotically stable for all .

Results

Existence of travelling waves for system (5)

In this section, we study the existence of travelling wave solutions to system (0.5) with nonlinear reaction terms satisfying .

We assume that a pair of upper-lower solutions and are given such that

(P1) ,

(P2) .

For the constants in , define by

(0.24)

(0.25)

(0.26)

The operators , and admit the following properties:

Lemma 0.1

Assume that and hold, then

for with , , .

Proof. By , direct calculation shows that

This completes the proof.

Lemma 0.2

([15]) Assume that and hold. Then for any , we have

(i) , .

(ii) and for with , , .

In terms of , and , system (0.34) can be rewritten as

(0.27)

Define

Let

and define by

for . It is easy to see that , and satisfy

(0.28)

Corresponding to Lemmas and , we have the following result for .

Lemma 0.3

Assume that and hold. For any , we have , , , , for with , , .

We next verify the continuity of .

Lemma 0.4

Assume holds, then is continuous with respective to the norm in .

Proof. For any fixed , let , then for with

direct calculation shows that

For , we see that

Similarly, for , we have

which implies that is continuous with respect to the norm in .

By using a similar argument as above, it can be shown that are continuous. Thus, we obtain that is continuous with respect to the norm in . This completes the proof.

Lemma 0.5

Assume that and hold, then

Proof. For any with , it follows from Lemma that

(0.29)

By the definition of upper-lower solutions, we have

(0.30)

Choosing in the first equation of (0.28), and denoting , we get

(0.31)

Setting and combining (0.30) and (0.31), we have

(0.32)

Repeating the proof of Lemma in Wu and Zou [15] shows that , which implies that .

By a similar argument, we know that , , , , , then . This completes the proof.

Lemma 0.6

Assume holds, then is compact.

Proof. Noting that

it follows from in Lemma that . By in Lemma and the fact that , we have

Hence, implies that there exists a constant such that .

For any ,

Thus, we have

If , we get

If , we obtain

Noting that , it follows from Lemma that is bounded by a positive number. Therefore, there exists a constant such that .

Similar to the proof of , we have that there exists a constant such that .

The above estimate for shows that is equicontinuous. It follows from the proof of Lemma that is uniformly bounded.

Next, we define

Then, for each , is also equicontinuous and uniformly bounded. Now, in the interval , it follows from Ascoli-Arzela Theorem that is compact. On the other hand, in as , since

By Proposition in [25], we have that is compact. This completes the proof.

Theorem 0.2

Assume that , and hold. Suppose there is a pair of upper-lower solutions and for (0.5) satisfying and . Then, system (0.5) has a travelling wave solution.

Proof. Combining Lemmas - with the Schauder's fixed point theorem, we know that there exists a fixed point of in , which gives a solution to (0.5).

In order to prove that the solution is a travelling wave solution, we need to verify the asymptotic boundary conditions (0.8).

By and the fact that

we get that

and

Therefore, the fixed point satisfies the asymptotic boundary conditions (0.8). This completes the proof.

Existence of travelling waves for system (0.3)

In this section, we use the results developed in Section 4 to study the existence of travelling wave solutions to system (0.3).

Denoting , then system (0.3) is equivalent to the following system

(0.33)

By making change of variables , , and dropping the tildes, system (0.33) becomes

(0.34)

It is easy to show that if , system (0.34) has two steady states and , where

On substituting , , and denote the travelling wave coordinate still by , we derive from (34) that

(0.35)

satisfying the following asymptotic boundary conditions

Lemma 0.7

The nonlinear reaction terms of system (0.34) satisfy .

Proof. For any , with , , , , we have

Let . We derive that .

For , it follows that

Let . We know that .

For , we have

Let . We obtain . This completes the proof.

Let

There exist such that

We can find that there exist satisfying

(0.36)

For the above constants, suitable constants and satisfying , and , we define the continuous functions and as follows

where is a constant to be chosen later. It is easy to know that , , , and satisfy (36) and the following conditions:

(P1) ,

(P2) .

Lemma 0.8

is an upper solution of system (0.35).

Proof. If , , and . It follows that

If , . We have

where

It follows from (0.36) that and there exists such that for all .

If , . It follows that

If , . We get

where

It follows from (0.36) that and there exists such that for all .

If , and . It follows that

If , . We get

where

It follows from (0.36) that and there exists such that for all .

Taking , we see that the conclusion is true. This completes the proof.

Lemma 0.9

is a lower solution of system (0.35).

Proof. If , . It follows that

If , and . We have

where

It follows from (0.36) that and there exists such that for all .

If , . It follows that

If , . We get

If , . We know

where

It follows from (0.36) that and there exists such that for all .

If , . It follows that

If , . We get

where

It follows from (0.36) that and there exists such that for all .

Letting , we see that our conclusion is true. This completes the proof.

Applying Lemmas -, we know that if , system (0.34) has a travelling wave solution with speed connecting the steady states and . Accordingly, we have the following conclusion.

Theorem 0.3

Let . For every , regardless of the value of , system (0.3) always has a travelling wave solution with speed connecting the uninfected steady state and the infected steady state .

Remark

The travelling wave solution in Theorem may not be monotonic. The fact is illustrated by the following numerical simulations.

Numerical simulations

In this section, by using the classical implicit format solving the partial differential equations and the method of steps for differential difference equations, we give the numerical simulations to illustrate the theoretical results above.

Example 1

In system (0.3), let , , , , , , , and . System (0.3) with above coefficients has a unique disease-free steady state . It is easy to show that the basic reproduction number of system (0.3) is . By Theorem we see that is locally stable. Numerical simulation illustrates our result (see Fig. 1).

Figure 1. The temporal solution found by numerical integration of problem (0.3) under conditions (0.4) and (0.15) with , , , , , , , , and , , , .

Example 2

In system (0.3), we set , , , , , , , and . System (0.3) with above coefficients has a disease-free steady state and an endemic steady state . It is easy to show that the basic reproduction number of system (0.3) is . Theorem shows that is unstable and is locally stable. It follows from Theorem that system (0.3) always has a travelling wave solution with speed connecting and . The fact is illustrated by the numerical simulation in Fig. 2. Clearly, Fig. 2 shows that the travelling wave solution does not possess the monotonicity, this seems to be due to the feature of the prey-predator interaction.

Figure 2. The temporal solution found by numerical integration of problem (0.3) under conditions (0.4) and (0.15) with , , , , , , , , and , , , .

Discussion

In this paper, we have dealt with the existence of travelling wave solutions for an SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. At first, by analyzing the corresponding characteristic equations, we discussed the local stability of a disease-free steady state and an endemic steady state to system (0.3) under homogeneous Neumann boundary conditions. We have shown in Theorem 0.1 that time delay and spatial diffusion are negligible for the local stability of the steady states to system (0.3). By using the cross iteration method and the Schauder's fixed point theorem, we reduced the existence of travelling wave solutions to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derived the existence of a travelling wave solution connecting the uninfected steady state and the endemic steady state .

In fact, we also find that time delay can influence the monotone of the travelling wave solution connecting the disease-free steady state and the endemic steady state . The effect of time delay on the travelling wave solution is illustrated by comparing the numerical simulations in Figs. 2 and 3.

Figure 3. The temporal solution found by numerical integration of problem (0.3) under conditions (0.4) and (0.15) with , , , , , , , , and , , , .